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(a) determine the domain, (b) sketch the graph and (c) determine the range of the function defined by the given equation.$g(x)=-2 x^{2}+4$

(a) $-\infty<x<\infty$(b)(c) $y \leq 4$

Algebra

Chapter 1

Functions and their Applications

Section 2

Basic Notions of Functions

Functions

McMaster University

Harvey Mudd College

Idaho State University

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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(a) determine the domain, …

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(a) find the domain of the…

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(a) identify the domain an…

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(a) draw the graph of the …

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for this problem, we've been given a function g of X equals negative two X squared plus four. There are three things we're going to do for this problem. We want to find the domain and the range for the function G, and then we're going to graph it. Sometimes visualizing the function helps to make sure that you have the correct domain and range given. So let's look at domain first domain are your inputs, those air your ex values? And when you're figuring out which values of X can go into the domain, it's often easier to look at this in reverse. Are there any excess that cannot go into the domain? Well, in this case, all I'm doing with the excess, I'm squaring it. You can square any real number. There's no limit to the exit. You could put in positive negative zero irrational. It doesn't matter. Since there is nothing that is excluded, the domain must include all real numbers. Now we can write this in two ways. I can write this as X is between negative infinity and positive infinity or I can write it in interval notation. Either one of these is valid now What about the range? My wives? Well, let's take a look at what we have here. X squared X squared is always positive. And then a multiplying it by a negative too. So this is always going to be negative at best is gonna be zero. That's the biggest it could never be for any other value of X. This is gonna be a negative number. So I'm gonna have four minus some number. The biggest y can be is four. Anything else? Any other value of X? Other the next equaling zero. This is just gonna get smaller and smaller and smaller. So my range is Why is less than or equal to four? If I wanted to write this an interval notation, I could say that this is a negative Infinity 24 And I'm going to include the four because, um, because I could I could let if x zero, it will actually equal four. So four is part of the range. So let's see what this looks like when we graph it. We already said, if x zero, why is four and why is less than that? So nothing is going to be up here in my graph, everything is going to go down. So let's pick a couple numbers. Let's let X equal one. Well, that means I'll have four minus two or two and I can. This is symmetric because Doesn't matter if X is positive one or negative one. When I square, it comes out to the same thing. What about effects is to or negative, too? That means I'll have four minus eight. Which means 123 I'll be down here it negative four. So if I connect thes, you can see that this is a fairly steep, um, downward parabola. You can see for sure that we got our range right. Everything was going down and my domain is correct. It's gonna take a long time. But I could get to any ex I wanted to if I made my graph large enough. So my domain range and sketch of the graph G of X

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